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Unformatted text preview: Math 235 Sample Term Test 1  1 Answers NOTE :  Only answers are provided here (and some proofs). On the test you must provide full and complete solutions to receive full marks. 1. Short Answer Problems a) Give the definition of an inner product h , i on a vector space V . Solution: h , i : V × V → R such that h ~v,~v i > 0 for all ~v 6 = ~ 0 and h ~ , ~ i = 0. h ~v, ~w i = h ~w,~v i h ~v,a~w + b~x i = a h ~v, ~w i + b h ~v,~x i b) Let B = { ~v 1 ,...,~v n } be orthonormal in an inner product space V and let ~v ∈ V such that ~v = a 1 ~v 1 + ··· + a n ~v n . Prove that a i = < ~v,~v i > . Solution: Taking the inner product of both sides with ~v i to get < ~v,~v i > = < a 1 ~v 1 + ··· + a n ~v n ,~v i > = a 1 < ~v 1 ,~v i > + ··· + a n < ~v n ,~v i > = a i since B is orthonormal. c) Define what it means for a set B to be orthonormal in an inner product space V . Solution: A set B = { ~v 1 ,...,~v n } is orthonormal in V if < ~v i ,~v j > = 0 for all i 6 = j and < ~v i ,~v i > = 1 for 1 ≤ i ≤ n ....
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This note was uploaded on 11/30/2010 for the course MATH 235/237 taught by Professor Wilkie during the Spring '10 term at Waterloo.
 Spring '10
 WILKIE
 Math

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